The expression \(\overline{\partial }_{A} {u}^\delta \)  can be expressed as a sum of
terms involving derivatives of the cutoff function \(\alpha \) , terms
involving derivatives of \(\xi _j\) , and terms involving the connection
\(A\)  on the bubble region. The derivative of \(\alpha \)  is bounded by \(C/ \rho | \delta |^{1/2}\) , while the norm of \(\xi _j\)  is bounded by
\(C\rho |\delta |^{1/2}\)  on the gluing region. Hence the term
involving the derivative of \(\alpha \)  is bounded and supported on a
region of area less than \( C|\delta | \rho ^2\) . In the given
trivialization we have
\( \overline{\partial }_A u = \overline{\partial }u + A_X^{0,1}(u) \) 
where \(A^{0,1}\)  is the \(0,1\) -form defined by \(A \in \Omega ^1(B_R,\mathfrak {g})\) 
is the connection 1-form in the local trivialization and
\(A_X^{0,1}(u)\)  is the corresponding form with values in \(T^{\operatorname{vert}}P(X) \otimes _\mathbb {R}. We have\)\( \Vert A_X^{0,1}(u) \Vert _{0,3,\delta } \le \Vert A_X^{0,1}(u)\Vert _{0,3} \)\(since \) p 2\(; for \) p = 2\( the \) W0,3,\( and \) W0,3\( normsare the same, by conformal invariance; for \) p > 2\( the \) 0,3\(-norm isstrictly greater. Hence\)\( \Vert \overline{\partial }_{A} u^\delta \Vert _{0,3,\delta } \le C\max (|\delta |^{1/3} \rho ^{2/3},|\delta |). \)\(The moment map term \) FA + (u)* P()  Vol\(vanishes except on \) |+| |1/2|\(, where it isuniformly bounded. Hence for \)\( small\)\( \Vert F_A + ({u}^\delta )^* P(\Phi )  \operatorname{Vol}_\Sigma \Vert _{0,3} \le C \rho ^{2/3} |\delta |^{1/3} .\)\(The statement of the lemma follows.\)
We also wish to perform the gluing construction in families, that is,
for each nearby vortex and gluing parameter we wish to find a solution
to the vortex equations on \(\Sigma ^\delta \) . Define
\( \mathcal {F}_{A,{u}}^{D,\delta } : \operatorname{Def}_\Gamma (A,{u}) \times \mathcal {I}_{{\delta }} \rightarrow \mathcal {H}_{{\delta }}, \quad (a,{\xi },a_1,\xi _1) \mapsto \\ \left( F_{A} + a_0+ a_1 +  \operatorname{Vol}_\Sigma \exp _{{u}^{{\delta }}}({\xi }_0^\delta +\xi _1)^* P(\Phi ), {\operatorname{d}}_{A,{u}^\delta }^* (a_0 +a_1,{\xi }_0^\delta + \xi ),\right. \\ \left. \Psi _{{u}^{{\delta }}}({\xi }_0^\delta + \xi _1)^{-1}\overline{\partial }_{A + a_0 + a_1}\exp _{{u}^{{\delta }}}({\xi }_0^\delta + \xi _1)\right) \) 
The following is proved in the same way as Lemma REF  and left
to the reader:
Lemma 4.2.5 
Let \((A,{u})\)  be as above. There exist constants \(c_0,c_1 > 0\) 
such that if \(|\delta | < c_1, \rho > 1/c_1 \) , \(\Vert (a,\xi ) \Vert \le c_1\)  and \(|\delta | \rho ^4 < c_1 \)  then
\(\Vert \mathcal {F}_{A,{u}}^{D,{\delta }}(a,\xi ,0,0) \Vert \le c_{0}|\delta |^{1/3} \rho ^{2/3}.\) 

Step 2: Uniformly bounded right inverse
In preparation for the construction of the uniformly bounded right
inverse of \(\tilde{D}_\delta \)  we define the intermediate family
\((A,{u}_0^\delta )\)  of gauged holomorphic maps on the nodal curve
\(\hat{\Sigma }\)  is the family defined by the equations (REF ),
using the identification of \(\hat{\Sigma }\)  and \(\hat{\Sigma }^\delta \) 
away from the gluing region. Thus \(u_0^\delta \)  is constant in a
neighborhood of the node \(w^\pm \) . We identify \(({u}_0^\delta )^*T^{\operatorname{vert}} P(X)\)  with \({u}^* T^{\operatorname{vert}} P(X)\)  by geodesic
parallel transport.
Lemma 4.2.6 
The operator \(\tilde{D}_{A,{u}_0^\delta }\) 
converges in the operator norm to \(\tilde{D}_{A,{u}}\)  as \(\delta \rightarrow 0, \rho \rightarrow \infty , \delta \rho ^4 \rightarrow 0\) .

The section \({u}_0^\delta \)  converges in the \(W^{1,3}\)  norm to \(u\) 
as \(\rho ^2 |\delta |^{1/2} \rightarrow 0\) . It follows that the operator \(\xi \mapsto  \operatorname{Vol}_\Sigma (u^\delta _0)^* L_\xi P(\Phi )\)  converges to \(\xi \mapsto  \operatorname{Vol}_\Sigma u^* L_\xi P(\Phi )\) . Hence \({\operatorname{d}}_{A,u_0^\delta ,\epsilon }\)  converges to \({\operatorname{d}}_{A,u,\epsilon }\) , and similarly
for \({\operatorname{d}}_{A,u_0,\epsilon }^*\) . The operator \( D_{A,{u}_0^\delta }\) 
converges to \(D_{A,{u}}\) , as in Lemma REF .
Proposition 4.2.7 
Let \((A,{u})\)  be a nodal vortex, and \((A,{u}^\delta )\)  the
approximate solution constructed above. There exist constants
\(c,C > 0\)  such that if \(| \delta | < c\)  then there exists an
approximate right inverse \(T_{\delta }\)  of the parametrized linear
operator \(\tilde{D}_{\delta } := \tilde{D}_{A,{u}^\delta }\)  that is,
a map \(T_{\delta } : \mathcal {H}_{\delta } \rightarrow \mathcal {I}_{\delta } \) 
such that
\( \Vert (\tilde{D}_{\delta } T_{\delta } - I) \eta \Vert _{\delta } \le {{2}}\Vert \eta \Vert _{\delta }, \quad \Vert T_{{\delta }} \eta \Vert _{\delta } \le C \Vert \eta \Vert _{\delta } .\) 

Given such an approximate inverse, we obtain a uniformly bounded right
inverse \(Q_{\delta }\)  to \(\tilde{D}_{\delta }\)  by the formula
\( Q_{\delta } = T_{\delta } (\tilde{D}_{\delta } T_{\delta })^{-1} = \sum _{k \ge 0}T_{\delta } (\tilde{D}_{\delta } T_{\delta } - I )^k .\) 
By the regularity assumption, \(\tilde{D}_{A,{u}} \)  is surjective when
restricted to the space of vectors \((a,{\xi })\)  such that \(\xi _0(0)= \xi _1(\infty )\) . By Lemma REF , \(\tilde{D}^0_\delta :=\tilde{D}_{A,{u}_0^\delta }\)  is surjective for sufficiently small
\(\rho ,\delta \) , when restricted to the same space. The approximate
right inverse is constructed by composing a cutoff operator
\(K_\delta \) , right inverse \(Q_\delta \) , and gluing operator \(R_\delta \) ,
as follows. (In other words, the right inverse is defined by
truncating the given functions, applying the right inverse for the
linearized operator on the nodal curve, and then gluing back together
using cutoff functions again.) For simplicity we assume that there is
a single node. Define the cutoff operator
\(K_{\delta }:\Omega ^{0,1}(\hat{\Sigma }^\delta ,{u}^{\delta ,*} T^{\operatorname{vert}}P(X))_{0,3,\delta }\rightarrow \Omega ^{0,1}(\hat{\Sigma },({u}^\delta _0)^{*}T^{\operatorname{vert}}P(X))_{0,3} \\K_\delta (\eta ) ={\left\lbrace \begin{array}{ll} 0 & \kappa _\pm ( z) \in B_{| \delta |^{{{2}}}} (0) \\\eta & \text{otherwise}\end{array}\right.} .\) 
Then \( \Vert K_\delta (\eta ) \Vert _{0,3} \le \Vert \eta \Vert _{0,3,\delta } .\)  Define the gluing operator
\( R_{\delta }:\Omega ^0(\hat{\Sigma },({u}^\delta _0)^{*} TX)_{{1,3}} \rightarrow \Omega ^0(\hat{\Sigma }^{\delta },u^{\delta ,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta }\) 
as follows. Let \(\hat{\Sigma }_\pm ^*\)  denote the complements of small
balls around the nodes \( \hat{\Sigma }_\pm ^* = \Sigma _\pm - B_{\rho ^2 |\delta |^{1/2}} (w^\pm )\) . Let \(\pi _\pm : \hat{\Sigma }_\pm ^* \rightarrow \hat{\Sigma }^\delta \)  denote the inclusions. These induce maps of
sections with compact support in \(\hat{\Sigma }_\pm ^*\) , \( \pi _{\pm ,*}:\, \Omega ^0_c( \hat{\Sigma }_\pm ^*,u_\pm ^* TX) \rightarrow \Omega ^0(\hat{\Sigma }^\delta , u^{\delta ,*} TX ) .\)  Define \(R_\delta (\xi ) = \xi ^\delta \)  where
\( \xi ^\delta = \pi _{+,*} \beta _{\rho ,\delta } (\xi _+ - \xi _+(w_+)) +\pi _{-,*} \beta _{\rho ,\delta } (\xi _- - \xi _-(w_-)) + \xi (w) \) 
for \( \kappa _\pm (z) \in B_{| \delta |^{1/2} \rho ^2} (0) \)  and \(\xi ^\delta = \xi \)  otherwise. Here \(x = \xi ^\pm (w^\pm )\)  is the value
of \(\xi \)  at the node. Define \( T_\delta := (I \times R_\delta )Q_\delta (I \times K_\delta ) .\)  That is, if \( (a,\xi ) = Q_\delta K_\delta (\phi ,\psi ,\eta ) \)  then \( T_\delta (\phi ,\psi ,\eta ) =(a,\xi ^\delta ) .\)  The map \(T_\delta \)  is the required approximate right
inverse. The difference \((\tilde{D}_\delta T_\delta -I)(\phi ,\psi ,\eta )\)  is the sum of terms
\( {\operatorname{d}}_{A,u^\delta } (a,\xi ^\delta ) - \phi , \quad {\operatorname{d}}_{A,u_0}^* (a,\xi ^\delta ) - \psi , \quad D_{A,u^\delta } (a,\xi ^\delta ) - \eta .\) 
By definition \({\operatorname{d}}_{A,u_0^\delta }(a,\xi ) = \phi \) , so the first
difference has contributions involving the difference between
\(u_0^\delta \)  and \(u^\delta \) , and between \(\xi ^\delta \)  and \(\xi \) .
But since \({\operatorname{d}}_{A,u}(a,\xi ) = {\operatorname{d}}_A a + u^* L_{\xi _X} P(\Phi )\) , these
terms are zeroth order in \(u,\xi \) ,
\( \Vert (u^\delta )^* L_{\xi ^\delta _X} P(\Phi ) - u_0^\delta L_{\xi _X}P(\Phi ) \Vert _{0,3} \le C | \delta | \Vert \xi \Vert _{C^0} ;\) 
that is, a constant times the area of \(\pi _{-,*} (\hat{\Sigma }_-^*)\) ,
which goes to zero as \(\delta \)  does. A similar discussion holds for
the second difference. The third difference has terms arising from
the cutoff function and the term \(a_X^{0,1}(u)\)  on the bubble region
\(|\kappa _+(z)| \le | \delta |^{1/2}\) .
We have
\( \Vert a_X^{0,1}(u) \Vert _{0,3,\delta } \le \Vert a_X^{0,1}(u) \Vert _{0,3} \le C |\delta | .\) 
since \(p \ge 2\) ; for \(p = 2\)  the \(W^{0,3,\delta }\)  and \(W^{0,3}\)  norms
are the same, by conformal invariance; for \(p > 2\)  the \(0,3\) -norm is
strictly greater. The term involving the derivative of the cutoff
function satisfies
\( \Vert {\operatorname{d}}\beta _{\rho ,\delta } (\xi - \xi (w)) \Vert _{0,3,\delta } \le c \log ( \rho ^2 )^{-2/3} \Vert (\xi - \xi (w)) \Vert _{1,3} \) 
by Lemma REF , and so vanishes in the limit \(\rho \rightarrow \infty \) . Using the uniform bound on \(Q_\delta \) , the total difference
is bounded by \(C ( \log ( \rho ^{2})^{-2/3} + |\delta |) \Vert (\phi ,\psi ,\eta ) \Vert \) , and so vanishes in the limit \(\delta \rightarrow 0,\rho \rightarrow \infty , |\delta | \rho ^2 \rightarrow 0\) . The uniform bound on \(T_{\delta }\) 
follows from the uniform bound on \(Q_\delta \)  and the cutoff and
extension operators.
Step 3: Uniform quadratic estimate
Proposition 4.2.8 
There exist constants \(c,C > 0\)  such that if \(\Vert \xi \Vert _{1,3,\delta } < c\) , \( |\delta | < c, \rho > 1/c\)  and \( |\delta |\rho ^4 < c \)  then the map \(\mathcal {F}^\delta _{A,{u}}\)  satisfies a
quadratic bound
\( \Vert D \mathcal {F}_{A,{u}}^\delta (a_1,\xi _1) -\tilde{D}_{A,{u}^\delta }(a_1,\xi _1) \Vert _{\delta } \le C \Vert a,\xi \Vert _{\delta } \Vert a_1, \xi _1 \Vert _{\delta } .\) 

The norm of the non-linear part of the curvature \(\Vert [a,a_1]\Vert _{{0,3}}\)  is bounded by Sobolev multiplication. The other term
appearing in the first vortex equation satisfies
\( \Vert \exp _{{u}^\delta } (\xi _0)^* L_{\xi _{1,X}}P(\Phi ) - ({u}^\delta )^* L_{\xi _{1,X}}P(\Phi ) \Vert _{{0,3}} \le C \Vert \xi _0 \Vert _{1,3,\delta } \Vert \xi _1 \Vert _{1,3,\delta } \) 
for some constant \(C\)  independent of \(\delta \) , using that
\(W^{1,3,\delta }\)  norm controls the \(W^{0,3}\)  norm uniformly. The
non-linear terms in the Cauchy-Riemann equation are estimated as in
Theorem REF  and [1]}; note
that we are fixing the complex structure on \(\Sigma \) , which avoids the
more complicated analysis we gave in the previous section. The second
vortex equation also involves a term of mixed type \( \Psi _{u}(\xi +\xi _1)^{-1} (a_1)_X^{0,1}(\exp _u(\xi + \xi _1)) - \Psi _u(\xi )^{-1}(a_1)_X^{0,1}(\exp _u(\xi )) .\)  It follows from uniform Sobolev
embedding that this difference has \(0,3,\delta \) -norm bounded by \(C\Vert a_1 \Vert _{{1,3}} \Vert \xi _1 \Vert _{1,3,\delta }\)  for some
constant \(C\)  independent of \(\delta \) .
Step 4: Implicit Function Theorem
Theorem 4.2.9  Let \((A,{u})\)  be a regular stable
nodal vortex of combinatorial type \(\Gamma \) . There exist constants
\(\epsilon _0,\epsilon _1 > 0\)  such that for every \((a,{\xi }, \delta ) \in \operatorname{Def}_\Gamma (A,{u})\)  with norm less than \(\epsilon _0\) , there exists
a unique \((\phi ,\psi ,\eta )\)  of norm less than \(\epsilon _1\)  such that if
\((a_1,\xi _1) = Q_\delta (\phi ,\psi ,\eta )\)  then \((A + a_0 + a_1,\exp _{{u}^{\delta }}({\xi }_0^{\delta } + \xi _1))\)  is a
symplectic vortex in Coulomb gauge with respect to
\((A,{u}^\delta )\) . The solution depends smoothly on
\(a_0,{\xi }_0\) , and transforms equivariantly the action of
\(\mathcal {G}(P)_{A,{u}}\)  on \(\operatorname{Def}_\Gamma (A,{u})\) .

Uniform error and quadratic estimates are those for
\(\mathcal {F}_{A,{u}}^\delta \)  in Lemmas REF , REF , and
REF , in a uniformly bounded neighborhood of 0 in
\(\operatorname{Def}_\Gamma (A,{u})\) . Then the first claim is an application of
the quantitative version of the implicit function theorem (see for
example [1]}). Equivariance follows from
uniqueness of the solution given by the implicit function theorem,
since the map \(\mathcal {F}^{D,\delta }_{A,{u}}\)  is equivariant for the
action of \(\mathcal {G}(P)_{A,{u}}\) .
Step 5: Rigidification
As in the case of holomorphic maps in the previous section, there is a
more natural way of parametrizing nearby symplectic vortices which
involves examining the intersections of the sections with submanifolds
of \(P(X)\) , and framings induced by parallel transport. First we study
the differentiability of the evaluation maps. The gluing construction
of the previous step gives rise to a deformation \((A_S,{u}_S)\)  of
\((A,{u})\)  with parameter space a neighborhood \(S\)  of 0 in
\(\operatorname{Def}(A,{u})\) , and so a map
\( S \rightarrow \overline{M}_n(\Sigma ,X), \ s \mapsto (\hat{\Sigma }_s,A_s,{u}_s) \) 
Consider the map
\( \operatorname{ev}: (\hat{\Sigma } - U) \times S \rightarrow P(X), \quad (z,s) \mapsto {u}_s(z) .\) 
Proposition 4.2.10 
The map \(\operatorname{ev}\)  of (REF ) is \(C^1\)  for the family constructed by
gluing in Theorem REF  using the exponential gluing profile.

We denote by \({u}_S^{\operatorname{pre}}: \hat{\Sigma }_S \rightarrow X\) 
the family obtained by pre-gluing only, that is, omitting the step
which solves for an exact solution. We denote by \(\operatorname{ev}^{\operatorname{pre}}\)  the map
\( \operatorname{ev}^{\operatorname{pre}}: (\hat{\Sigma } - U) \times S \rightarrow P(X), \quad (z,s) \mapsto {u}_s^{\operatorname{pre}}(z) .\) 
This map is independent of the gluing parameters, and is therefore
\(C^1\) . We write \(s = (a_0,\xi _0)\)  and \(A_s = A + a_0 + a_1, {u}_s= \exp _{{u}^{{\operatorname{pre}}}_s}(\xi _0^\delta + \xi _1)\) . The corrections
\(a_1,\xi _1\)  depend smoothly on \(a_0,\xi _0\) , by the implicit function
theorem, and so \(\xi _1(z)\)  depends smoothly on \(a_0,\xi _0\) . Next we
take the derivative with respect to the gluing parameter. Let
\((A,{u})\)  be a nodal symplectic vortex, \((A,{u}^\delta )\)  the
pre-glued pair (we omit the parameter \(\rho \)  controlling the diameter
of the gluing region from the notation) and consider the equation \(\mathcal {F}_{A,{u}^\delta } (a_0 + a_1, \xi _0^\delta + \xi _1) = 0 .\) 
Let \(\tilde{D}_\delta \)  denote the derivative of \(\mathcal {F}_{A,{u}^\delta }\) .
Differentiating with respect to \(\delta \)  gives
\( \tilde{D}_\delta \left(\frac{d}{d \delta }a_1 , D\exp _{{u}^\delta }(\xi _0^\delta ;0 , \frac{d}{d \delta }\xi _1 )\right) = - \tilde{D}_\delta \left(0,D\exp _{{u}^\delta }(\xi _0^\delta ; \frac{d}{d \delta }{u}^\delta , \frac{d}{d \delta }\xi _0^\delta )\right) .\) 
The same arguments as in the proof of Theorem REF  show that
there exists a constant \(C > 0 \)  such that the right hand side is
bounded in norm by \(C e^{-1/\delta }\) . On the other hand, the norm of
the left-hand side \(\tilde{D}_\delta \)  is uniformly bounded from below in
terms of the norm of \(\frac{d}{d \delta }a_1, \frac{d}{d \delta }\xi _1\) , by the quadratic
estimates. It follows that \((\frac{d}{d \delta }a_1, \frac{d}{d \delta }\xi _1)\)  is also bounded
in norm by \(C e^{-1/\delta }\) . Hence \( \lim _{\delta \rightarrow 0}\partial _\delta \operatorname{ev}= 0 \) . It follows that \(D\operatorname{ev}\)  has a continuous
limit as \({\delta } \rightarrow 0\) .
Choose a path \(\gamma : [0,1] \rightarrow \Sigma \)  in the principal component
and an element \(\phi _0 \in P_{\gamma (0)}\) . Let \(\tau _\gamma (A):P_{\gamma (0)} \rightarrow P_{\gamma (1)}\)  denote parallel transport. By an
\(m\) -framed family of marked curves, we mean a family of curves
together with an \(m\) -tuple of points in \(P\) . Given a family
\((\hat{\Sigma }_S,A_S,{u}_S)\)  of gauged holomorphic maps over a
parameter space \(S\) , a collection of codimension two submanifolds
\({Y} = (Y_1,\ldots ,Y_k)\)  in \(P(X)\) , and a collection of paths
\({\gamma } = (\gamma _1,\ldots , \gamma _l)\)  with the same initial
point \(y_0\)  to \(y_j, j = 1,\ldots , l\) , define a family of marked,
framed curves
\(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow S\) 
by requiring that the additional marked points \(z_{n+i}\)  map to \(Y_i\) ,
and the framings are given by parallel transport along the paths
\(\gamma _i\) .
Definition 4.2.11 
The data \({Y},{\gamma },A,{u}\)  are compatible if

each \(Y_j\)  intersects \(u_j\)  transversally in a single point \(z_j\in \hat{\Sigma }\) ;

if \((a,\xi ) \in  \operatorname{ker}\tilde{D}_{A,{u}}\)  satisfies \(\xi (z_{n+j})\in T_{{u}(z_{n+j})} P(X) \)  for \(j = 1,\ldots , k\)  and \(D_A\tau _{\gamma _i}(a) = 0 \)  for \(i = 1,\ldots , l\)  then \((a,\xi ) = 0\) .

the curve \(\hat{\Sigma }\)  marked with the additional points
\(z_{n+1},\ldots , z_{n+k}\)  is stable.

if some automorphism of \((\hat{\Sigma },{u})\)  maps \(z_i\)  to
\(z_j\)  then \(Y_i\)  is equal to \(Y_j\) .


The second condition says that there are no infinitesimal deformations
which do not change the positions of the extra markings or framings.
Proposition 4.2.12 
Let \((A,{u})\)  be a parametrized regular stable nodal vortex, and
\((A_S, {u}_S) \rightarrow S\)  the stratified-smooth universal deformation
constructed by the gluing construction. There exists a collection
\(({Y},{\gamma })\)  compatible with \(({u},A)\) . Furthermore, if
\(({Y},{\gamma })\)  is compatible with \((A,{u})\) , then the
rigidified family \(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow S\)  of (REF ) is a stratified-smooth deformation of the
marked-curve-with-framings
\(\hat{\Sigma }^{{Y},{u},{\gamma },A}\)  which defines an
immersion of \(S\)  into the parameter space for the universal
deformation of the central fiber.

First we show the existence of a compatible collection. Suppose that
the second condition is not satisfied for some \((a,\xi )\) . Suppose
first that \(\xi \ne 0\) . Let \(z_{n+1}\)  be a point with \(\xi (z_{n+1})\ne 0\) , and choose a codimension two submanifold \(Y_{n+1}\)  transverse
to \(u\)  near \(u(z_{n+1})\) , and such that \(TY_{n+1}\)  does not contain
\(\xi (z_{n+1})\) . Adding \(Y_{n+1}\)  to the list of submanifolds
decreases the dimension of the space of \((a,\xi )\)  satisfying the
condition in (b) by at least one. Repeating this process, we may
assume that the only elements satisfying the condition in (b) have
\(\xi =0 \) . Suppose that \(\xi \)  is zero, so that \(a\)  is necessarily
non-zero. Choose an additional marked point \(y_{l+1}\)  and a path
\(\gamma _{l+1}\)  from the base point \(y_0\)  to \(y_{l+1}\)  such that the
derivative of the parallel transport over \(\gamma \)  with respect to \(a\) 
over is non-zero. Appending \(\gamma _{l+1}\)  to the list of path
decreases the dimension of \((a,\xi )\)  satisfying the condition in (b)
by at least one. Hence the process stops after finitely many steps,
after which the kernel is trivial. The proof of the second claim is
similar to Proposition REF  and will be omitted.
Step 6: Surjectivity
We show that any nearby vortex appears in the family constructed
above. First, we show:
Proposition 4.2.13 
Let \((A,{u})\)  be a regular strongly stable symplectic vortex.
There exists a constant \(\epsilon > 0\)  such that if \((A_1,u_1) = (A + a,\exp _{{u}^\delta }(\xi ))\)  with \(\Vert a \Vert _{1,3} + \Vert \xi \Vert _{1,3,\delta } \le \epsilon \)  then after gauge transformation we have
\((A_1,u_1) = (A+ a_0 + a_1,\exp _{{u}^\delta }(\xi _0^\delta +\xi _1))\)  for some \((a_0,\xi _0) \in  \operatorname{ker}(\tilde{D}_{A,{u}})\)  and
\((a_1,\xi _1)\)  in the image of \(Q_\delta \) .

We claim that for some constant \(C > 0\) , we have \((a,\xi ) =(a_0,\xi _0^{{\delta }}) + (a_1,\xi _1)\)  for some \((a_0,\xi _0) \in  \operatorname{ker}\tilde{D}_{A,{u}}\)  and \((a_1,\xi _1) \in \operatorname{Im}\tilde{D}_{A,{u}^{{\delta }}}^*\)  with norm \( \Vert (a_1,\xi _1) \Vert \le C \Vert (a_1,\xi _1) \Vert .\)  Given the claim, the proposition
follows by the uniqueness statement of the implicit function theorem.
For any \(c > 0\)  there exists \(\delta _0\)  such that for \(\delta <\delta _0\) ,
\( \Vert \tilde{D}_{A,{u}^{{\delta }}} (a_0^{{\delta }},{\xi }_0^{{\delta }}) \Vert \le c \Vert (a_0^{{\delta }},{\xi }_0^{{\delta }}) \Vert \) 
by estimates similar to those of Lemma REF . Thus the image
of \( \operatorname{ker}\tilde{D}_{A,{u}}\)  is transverse to \(\operatorname{Im}\tilde{D}_{A,{u}^\delta }^*\) , for \(\delta \)  sufficiently small, since
it meets \(\operatorname{Im}\tilde{D}_{A,{u}^\delta }^*\)  trivially and projects
isomorphically onto \( \operatorname{ker}\tilde{D}_{A,{u}^\delta }\) , by gluing for
indices, as in [3]}. The claim then follows
from the inverse function theorem.
Given a strongly stable symplectic vortex \((A,{u})\)  with stable
domain \(\hat{\Sigma }\) , let \((A_S,{u}_S)\)  be the family given by
the gluing construction above. Otherwise, if \({Y}\)  is not stable,
let \({Y} = (Y_1,\ldots , Y_l)\)  be a collection of codimension two
submanifolds of \(P(X)\) , and consider the family
\((A,{u}^{{Y}})\)  with additional marked points given by
requiring that the additional marked points \(z_{n+i}\)  map to \(Y_i\) .
Let \((A_S,{u}_S)\)  denote the family obtained by applying the
gluing construction for \((A_S,{u}_S^{{Y}})\) , and then
forgetting the additional marked points.
Lemma 4.2.14 Suppose that \((A_i,u_i)\)  Gromov converges to \((A,u)\) .
After a sequence of gauge transformations, for any \(\epsilon \) , there
exists \(i_0\)  such that if \(i > i_0\)  then there exists
\(\delta ,(a_i,\xi _i)\)  satisfying \((A_i,u_i) = (A + a_i,\exp _{{u}^\delta }(\xi _i))\)  with \(\Vert a_i \Vert _{1,3} + \Vert \xi _i \Vert _{1,3,\delta } \le \epsilon \) .

By definition of Gromov convergence, after gauge transformation \(A_i\)  \(C^0\) -converges to \(A\) 
and converges uniformly in all derivatives on the complement of the
bubbling set [4]}. The exponential decay estimate
[4]} show that \(u_i\)  converges to \(u\)  on the
complement of the nodes, uniformly in all derivatives on compact sets,
and whose derivative on the gluing region is uniformly bounded in the
\(\delta \) -dependent metric. It follows that \(u_i =\exp _{u^\delta }(\xi _i)\)  for some \(\delta \)  and \(\xi _i \in \Omega ^0(\Sigma ^\delta , (u^\delta ) T^{\operatorname{vert}} P(X))\)  with \(\Vert \xi _i \Vert _{1,3,\delta } < \epsilon \) . To obtain the improved convergence
for the connection, note that \( F_{A_i} + (u_i)^* P(\Phi ) = 0 \)  and
the corresponding equations for the limit \((A,u)\)  imply that
\( F_{A_i} - F_A = {\operatorname{d}}_A (A_i - A) - (A - A_i) \wedge (A - A_i) =(u_i)^* P(\Phi ) - u^* P(\Phi ) .\) 
Since \(u_i^* P(\Phi )\)  is bounded and converges to \(u^* P(\Phi )\)  on the
complement of the bubbling set, and \(A_i\)  converges to \(A\)  in \(C^0\) 
hence \(W^{0,3}\) , the right hand side converges to 0 in \(W^{0,3}\)  as
\(i \rightarrow \infty .\)  After gauge transformation we may assume that \({\operatorname{d}}_A^* (A - A_i) = 0\) . Then the elliptic estimate for the operator
\({\operatorname{d}}_A \oplus {\operatorname{d}}_A^*\)  implies that \(A - A_i\)  converges to zero in
\(W^{1,3}\) .
Corollary 4.2.15 
\((A_S,{u}_S)\)  is a stratified-smooth versal deformation of
\((\hat{\Sigma },A,{u})\) .

Proposition REF  implies that any family
\((A_{S^1}^1,{u}_{S^1}^1)\)  is obtained by pull-back from
\((A_S,{u}_S)\) , in case \(\hat{\Sigma }\)  is stable, or obtained from
the family obtained by adding the marked points mapping to
submanifolds, in general.
Step 7: Injectivity
We show that any nearby vortex appears once in our family, up to the
action of \( \operatorname{Aut} (A,u)\) ; this is part of the following:
Theorem 4.2.16 Any family \((A_S,{u}_S)\)  constructed
by gluing using the exponential gluing profile is a strongly universal
stratified-smooth deformation of \((A,{u})\) .

Let \(\overline{Z}_n(P,X)\)  denote the moduli space of marked symplectic
vortices up to equivalences that involve only the identity gauge
transformation, so that \( \overline{M}_n(P,X) = \overline{Z}_n(P,X) / \mathcal {G}(P) .\)  Let
\((A,{u})\)  be a stable marked vortex, and \(W_{A,{u}}\)  a slice
for the gauge group action on \(\overline{Z}(P,X)\) , so that
\(W_{A,{u}}/\mathcal {G}(P)_{A,{u}} \rightarrow \overline{M}_n(P,X) \) 
is a homeomorphism onto its image. Let \( \operatorname{Aut} _0(A,{u})\)  denote the
subgroup of \( \operatorname{Aut} (A,u)\)  acting trivially on \(P\) , so that
\(\mathcal {G}(P)_{A,{u}} =  \operatorname{Aut} (A,u)/ \operatorname{Aut} _0(A,{u})\)  is the stabilizer of
\((A,{u})\)  under the gauge action. Let \((A_S,{u}_S)\)  denote a
universal deformation of \((A,{u})\)  constructed by gluing using the
exponential gluing profile. We claim that the map
\(  S/ \operatorname{Aut} _0(A,{u}) \rightarrow W_{A,{u}}, \quad [s]\mapsto [A_s,{u}_s] \) 
is an injection. Indeed, rigidification produces an injection
\(  S/ \operatorname{Aut} _0(A,{u})\rightarrow \overline{M}_{n+k,l}(\Sigma )/ \operatorname{Aut} _0(A,{u}), \quad [s] \mapsto [\Sigma ^{A_s,{u}_s,{Y},{\gamma }}] \) 
where \( \operatorname{Aut} _0(A,{u})\)  acts by re-ordering the marked points.
Since this map factors through (REF ), the claim follows. If
\((A_{S^1},{u}_{S^1}^1)\)  is a family of symplectic vortices giving
a deformation of any fiber of \((A_S,{u}_S)\) < then Corollary
REF  together with injectivity shows that this family is
obtained by pull-back by some map \(S^1 \rightarrow S\) . Hence
\((A_S,{u}_S)\)  is a stratified-smooth strongly universal
deformation of \((A,{u})\) .
Theorem 4.2.17  Let \(X\)  be a Hamiltonian \(G\) -manifold
equipped with a compatible invariant almost complex structure \(J \in J(X)^G\) . The maps
\( S \rightarrow \overline{M}_n(\Sigma ,X), \quad s\mapsto [A_s,{u}_s] \) 
associated to the universal deformations constructed above equip the
locus \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X)\)  of regular stable symplectic vortices
with the structure of a stratified-smooth orbifold. If the local
coordinates near the nodes are chosen compatibly and the gluing
profile is the exponential gluing profile, then the deformations
provide \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X)\)  with the structure of a
\(C^1\) -orbifold.

It suffices to show that the charts given by two sets
\({Y}_j,{\gamma }_j\)  are compatible. Define \({Y} ={Y}_1 \cup {Y}_2\)  and \(m = m_1 + m_2\)  the total number of
extra points. Similarly let \({\gamma }\)  be the union of
\({\gamma }_1\)  and \({\gamma _2}\)  of total number \(l = l_1 +l_2\) . The family \( \hat{\Sigma }^{{Y},{u},{\gamma },A}_S\) 
admits a proper étale forgetful map \(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow \hat{\Sigma }_S^{{Y}_j,{u},{\gamma }_{j},A}, \quad j = 1,2\) 
whose fiber consists of the re-orderings of the points for \({Y}\) 
induced by automorphisms of \( \operatorname{Aut} (A,{u})\)  that fix the ordering
for \({Y}_j\) . It follows that the corresponding charts are
\(C^1\) -compatible.
Remark 4.2.18 As discussed in Remark REF , the Theorem
implies that if \(\overline{M}^{{\operatorname{reg}}}_n(\Sigma ,X)\)  is compact then it admits
a (non-canonical) smooth structure.

Let \(\overline{M}_n(\Sigma )\)  denote the moduli space of stable maps to
\(\Sigma \)  with homology class \([\Sigma ]\) , \(n\)  markings and genus that
of \(\Sigma \) , or in other words, parametrized stable curves with
principal component isomorphic to \(\Sigma \) . Forgetting the pair
\((A,u)\)  gives a forgetful morphism \( \overline{M}^{{\operatorname{reg}}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma ) .\)  Using the differentiable structure defined above,
the evaluation maps are differentiable but unfortunately the
forgetful morphisms are not, unless one uses a different gluing
profile for the moduli space of vortices with one less marking. More
precisely, the forgetful morphism \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma )\)  is continuous and \(C^1\)  near any pair \((A,{u})\) 
whose domain is stable as an element of \(\overline{M}_n(\Sigma )\) , and a
submersion near the boundary of \(\overline{M}_n(\Sigma )\) . For the standard
smooth structure on \(\overline{M}_n(\Sigma )\) , the forgetful morphism
\(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma )\)  is smooth.
The gluing construction has various parametrized versions. For
example, in [6]} we consider a moduli space of polystable
polarized vortices, which consist of a vortex together with a lift
of the connection to the Chern-Simons line bundle. In each of these
cases one applies the implicit function theorem using the linearized
operator for the parametrized problem to prove that any parametrized regular polystable vortex has a strongly universal
deformation in the parametrized sense. In particular, any regular
polystable polarized vortex has a strongly universal deformation etc.
